arXiv subjects

Congming Li

Publications and source records attributed to Congming Li.

40 records · Page 3Linked to original sources

arXiv2009

Global Regularity of the 3D Axi-symmetric Navier-Stokes Equations with Anisotropic Data

In this paper, we study the 3D axi-symmetric Navier-Stokes Equations with swirl. We prove the global regularity of the 3D Navier-Stokes equations for a family of large anisotropic initial data. Moreover, we obtain a global bound of the solution in terms of its initial data in some $L^p$ norm. Our results also reveal some interesting dynamic growth behavior of the solution due to the interaction between the angular velocity and the angular vorticity fields.

math.AP↗

arXiv2007

Uniqueness of positive bound states to Schrodinger systems with critical exponents

We prove the uniqueness for the positive solutions of the following elliptic systems: \begin{eqnarray*} \left\{\begin{array}{ll} - \lap (u(x)) = u(x)^αv(x)^β - \lap (v(x)) = u(x)^β v(x)^α \end{array} \right. \end{eqnarray*} Here $x\in R^n$, $n\geq 3$, and $1\leq α, β\leq \frac{n+2}{n-2}$ with $α+β=\frac{n+2}{n-2}$. In the special case when $n=3$ and $α=2, β=3$, the systems come from the stationary Schrodinger system with critical exponents for Bose-Einstein condensate. As a key step, we prove the radial symmetry of the positive solutions to the elliptic system above with critical exponents.

math.AP↗

arXiv2006

Dynamic Stability of the 3D Axi-symmetric Navier-Stokes Equations with Swirl

In this paper, we study the dynamic stability of the 3D axisymmetric Navier-Stokes Equations with swirl. To this purpose, we propose a new one-dimensional (1D) model which approximates the Navier-Stokes equations along the symmetry axis. An important property of this 1D model is that one can construct from its solutions a family of exact solutions of the 3D Navier-Stokes equations. The nonlinear structure of the 1D model has some very interesting properties. On one hand, it can lead to tremendous dynamic growth of the solution within a short time. On the other hand, it has a surprising dynamic depletion mechanism that prevents the solution from blowing up in finite time. By exploiting this special nonlinear structure, we prove the global regularity of the 3D Navier-Stokes equations for a family of initial data, whose solutions can lead to large dynamic growth, but yet have global smooth solutions.

math.AP↗

arXiv2003

Qualitative Properties of Solutions for an Integral Equation

Let $n$ be a positive integer and let $0 < α< n.$ In this paper, we continue our study of the integral equation $$ u(x) = \int_{R^{n}} \frac{u(y)^{(n+α)(n-α)}{|x - y|^{n-α}}dy.$$ We mainly consider singular solutions in subcritical, critical, and super critical cases, and obtain qualitative properties, such as radial symmetry, monotonicity, and upper bounds for the solutions.

math.AP↗